Thomason condition for regular algebraic stacks

TL;DR

This paper proves that immersions into concentrated regular Noetherian algebraic stacks with quasi-finite and locally separated diagonals satisfy the $1$-Thomason condition, yielding a singly generated derived category of quasi-coherent sheaves on the immersion. The authors develop a two-step strategy: descent along finite flat presentations to reduce to étale cases, and a stacky recollement to glue compact generators across open-closed decompositions. This extends Hall–Rydh's results from quasi-finite separated diagonals to locally separated diagonals under regularity and concentration assumptions, producing new cases including non-separated diagonals and positive characteristic. As a consequence, $D(\operatorname{Qcoh}(\mathcal{X}))$ is equivalent to $D_{\operatorname{qc}}(\mathcal{X})$ for such stacks, and the Balmer spectrum is determined by the underlying topological space, enhancing the toolkit for derived-category and K-theory applications in algebraic geometry.

Abstract

We show that any immersion into a concentrated regular Noetherian algebraic stack with quasi-finite and locally separated diagonal satisfies the Thomason condition. In particular, the derived category of quasi-coherent sheaves on such an algebraic stack is singly compactly generated. This extends results of Hall--Rydh from separated to locally separated diagonal, under additional regularity hypotheses.

Theorems & Definitions (16)

• Theorem 1.1: see \ref{['thm:thomason']}
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Proposition 4.3
• proof